Functional Quantum Calculus Approach to Degree-Based Topological Indices in Chemical Graph Theory

In this research article, we replace the integer degree n = deg(v) of a vertex by its q(t)-analogue [n]q(t) and by an h(t)-analogue ⟨n⟩h(t) obtained from the h(t)-derivative of x n at x = 1, and we show that every classical degree-based index TI(G) = ∑︁uv∈E(G) F(deg u, deg v) admits two functional companions TIq(t)(G) and TIh(t)(G) that recover TI(G) exactly in the limit t → 0. We establish convergence, monotonicity, boundedness and a functional congruence theorem for these deformed indices, specialise the construction to the first and second Zagreb indices, the Randi´c index, the ABC index, the GA index, the harmonic index and the Sombor index, and verify all theoretical claims numerically on path, cycle, star and fused-ring (naphthalene) molecular graphs. We then give an explicit chemical application, showing how an edge-weighted extension of the same framework tracks a degreebased topological index continuously along the reaction coordinate of a Diels–Alder cycloaddition, recovering the classical reactant- and product-graph indices at the two endpoints.